We have no reason to believe the bound is sharp. The Kodaira-Parshin trick (as
used in section:counterexample, for example) is one source of
variations of Hodge structure on \({M}_{g,n}\)  of rank bounded in terms of
\(g, n\) , but it is not the only one. For example, the representations constructed
in [1]} are cohomologically rigid and hence underlie
integral variations of Hodge structure by [2]} and [3]}.
Assuming Simpson's motivicity conjecture ([3]}) these constructions are geometric in nature, though this is not clear from the construction. Of course it would be extremely interesting to prove that these representations arise from algebraic geometry.
